Square root relationship

Thus, if we knew the second moment of the local density of states we should be able to determine the atomic binding energy via the square root relationship. However, as quantum... 

The wedge design maintains a square root relationship between flow rate and differential pressure for pipe Reynolds numbers as low as approximately 500. The meter can be flow caUbrated to accuracies of approximately 1% of actual flow rate. Accuracy without flow caUbration is about 5%. 

La.mina.r Flow Elements. Each of the previously discussed differential-pressure meters exhibits a square root relationship between differential pressure and flow there is one type that does not. Laminar flow meters use a series of capillary tubes, roUed metal, or sintered elements to divide the flow conduit into innumerable small passages. These passages are made small enough that the Reynolds number in each is kept below 2000 for all operating conditions. Under these conditions, the pressure drop is a measure of the viscous drag and is linear with flow rate as shown by the PoiseuiHe equation for capilary flow ... 

The total flow rate, (Lmn + Gmn), leaving the mixer will be related to the total phase volumes VLm and Vcm by a hydrostatic equation, which will depend on the net difference in the head of liquid, between the levels in the mixer and in the settler. The actual form of this relationship might need to be determined experimentally, but could, for example, follow a simple, square-root relationship of the form, in which flow rate is proportional to the square root of the difference in liquid head, or indeed to the total volume of liquid in the mixer, e.g.. 

An experimental test demonstrated the validity of the square root relationship. Experimental conditions affecting data resulting from this device include stirring rate, temperature control, and sink conditions. 

Another specialized form of voltammetry involves the use of either a rotated-disk or a ring-disk indicating electrode. With this type of electrode the current is directly proportional to the square root of the rate of rotation if it is a diffusion-controlled process. To obtain complete adherence to the square-root relationship, a hydrodynamically sound design for the electrode is essential.43 Figure 3.12 illustrates the geometric features that have been found to give reliable performance for rotation rates as high as 10,000 rpm. 

The V-cone flowmeter requires less upstream straight pipe and maintains the square root relationship between flow and pressure drop at lower Reynolds numbers than does an orifice plate. (Courtesy of McCrometer Div. of Danaher Corporation.)... 

A log-log plot of ionization signal versus laser power, as shown in Fig. 4, shows an initial slope of 0.5 and tends to level off at higher densities. This again can be explained by the equilibrated process (1), which leads to the square root relationship between ion-density and excited state density for low laser power where no saturation occurs. The initial slope of 0.5 also shows that multiphoton ionization is negligible since the latter process would give rise to a stronger dependency of ionization signal on the laser power. 

Figure 5.8 Inverse square root relationship between strength and diameter of a patented, pearlitic steel wire (after Embury and Fisher, 1966).

In this regard, a useful feature of many spectrometers is the block-size parameter. For NMR, and in general, block sizes are set to 4 or a multiple of 4, again depending on the concentration of the sample. The number of scans is set to an accumulation time that might correspond to the total time that the operator has reserved. (Modern spectrometers also have programs that calculate the total experimental time from the number of scans, and any delay times.) At the end of each block, the summed FIDs (see earlier) are written into the computer memory, where they can be Fourier transformed. When the spectrum displays a sufficient S/N ratio, the acquisition is halted. Remember, from the square-root relationship just discussed, in order to double the S/N ratio, we must quadruple ns. 

It is believed that the free positive and negative species annihilate each other immediately on contact. This is borne out by the strict square root relationship which is found between the rates of polymerization and the dose rate of the radiation. The growing chain ends are therefore free in nature, i.e., with no ion-pair component. This makes radiation initiated ionic polymerization an excellent method for studying free ion polymerization examples of the power of this method have been presented for p-methoxy styrene (1 9) and the vinyl ethers (16,20,21). 

DP Werthemann. Why Redox Pulping Catalysts Pit the Square Root Relationship. 

These generalizations may be illustrated by example (61). Consider the experiments outlined in Table 1. PHEMA, PMMA and copolymers of the two were prepared by dilute solution polymerization. Copolymer compositions (Table 1A [parentheses] ), obtained by application of proton Nuclear Magnetic Resonance spectroscopy (as detailed later in the discussion), are almost identical to the monomer feed ratios, despite high conversions. The latter increase through series 3 to 1, due to the square root relationship between initiator concentration and instantaneous rate of polymerization (63). Close to random co-polymerizations have been observed also in monomer mixtures containing... 

HEMA, MMA, and MAA or DMAEMA (66, 67). Polymer molecular weight control was attempted by adjustment of initiator concentrator through the presumed inverse square root relationship between initiator level and kinetic chain length, (63) and this was reflected in measured intrinsic viscosities. Comparisons across a series are made difficult by changing Mark-Houwink-Sakurada coefficients. However direct comparisons may be made between polymers of similar composition, for example lc, 2c, and 3c, to show an expected increase in hydrodynamic volume from series 1 through series 3. 

Equation (3.3) comes from the square-root relationship between velocity and pressure drop. Dropping the flow by a factor of 2 cuts the AP signal by a factor of 4. For system analysis we usually linearize Eq. (3.3) around the steady-state value of flow rate, F. 

The last three systems are best applied well before corrosion has initiated. As the chloride profile within the concrete approximates to a square root relationship, there will be considerable reserves of chloride in the cover concrete to push the concentration at rebar level above the corrosion... 

For pyritic sulfur oxidation, various dependence of rate on partial pressure of oxygen and pyritic sulfur concentration are reported in the literature. Within the temperature range of 100-130°C and oxygen partial pressures up to 0.4 MPa, McKay and Hal-pern (53) reported the rate of pyritic sulfur oxidation to be first order in oxygen partial pressure and zero order with respect to pyritic sulfur. A kinetic study of the pyrite oxidation at Kennecott Copper Corporation (58) indicated a square root relationship between oxygen partial pressure and the rate of pyrite oxidation with an activation energy of 58.6 kJ/mole for the rate constant. 

Pooling all of the data taken at the two flow rates provides a relation between TCE conversion and UV intensity (Figure 22.4.6). Significant scatter, which exists for both flow rates, is likely due to the rapidly varying UV intensity levels during cloud events. The variable illumination makes it difficult to establish a precise correlation between an isolated UV measurement and the U V exposure received by the gas as it flows through and reacts with the photocatalyst bed. The 10 SCFM data suggests a square root relationship between UV intensity and conversion. The square root dependence was observed in a continuous flow reactor when TCE concentrations were less than 60 ppmv (Nimlos et. al., 1993). Hi er concentrations... 

Based on the above scheme, Bowman et al., predicted that the rate might be proportional to the square root of the product of monomer eoneentration and rate of initiation. Cook and Patisson, on the other hand, claim that their investigation shows that influence of irradiation intensity and initiator eoneentration on the cure rate does not follow the classical square root relationship. 

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